Astrophysical S(E)-factor of the 15N(p,α)12C reaction at sub-Coulomb energies via the Trojan-horse...
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Astrophysical S(E)-factor of the 15N(p,α)12C reaction at sub- Coulomb energies via the Trojan- horse method Daniel Schmidt, Liberty University Cyclotron Institute —Texas A&M University Mentor: Dr. Akram Zhanov Background The CNO Cycle in Stellar Nucleosynthesis 12 C + p → 13 N + γ 13 N → 13 C + e + + ν 13 C + p → 14 N + γ 14 N + p → 15 O + γ 15 O → 15 N + e + + ν 15 N + p → 12 C + α In this branch of the CNO (carbon, nitrogen, oxygen) cycle, 12 C acts as a nuclear catalyst which allows the fusion of four protons into a 4 He nucleus (or α particle). The final step of the CNO cycle, 15 N + p → 12 C + α (also written 15 N(p,α) 12 C) is the subject of this research. When a proton impinges on the 15 N nucleus, it is repelled by the Coulomb force from the seven other protons, and thus it must have sufficient energy to overcome this “Coulomb barrier” of 2.14 MeV if it is to react. At energies typical of stellar interiors, protons have energies less than 100 keV. The reaction still occurs, but it depends on quantum tunneling. Accelerator experiments can reproduce the 15 N(p,α) 12 C reaction at E> 70 keV, but at astrophysically relevant energies, the reaction proceeds too slowly to be measured efficiently in this way. The Trojan Horse Method An indirect technique, the Trojan horse method, can be used at low energies. In this method, a deuteron impinges on a 15 N nucleus. After penetrating the Coulomb barrier, the deuteron breaks, dividing its energy between the proton and neutron. (See diagram below.) In some cases, the neutron will exit the scene with most of the deuteron’s energy, leaving a low- energy proton inside the Coulomb barrier to react with the 15 N nucleus. By selecting those events in which the neutron leaves with high energy, we can study the reaction at low energies without suppression from the Coulomb barrier. 15 N α 12 C n 16 O p d The next task is to see if we can find a theoretical correction to the Trojan horse data that will bring the two methods into agreement. Differences between the two methods include the following: – In the direct method, the proton is affected by the Coulomb barrier, while in the Trojan horse method it is not. – The direct data is also affected by a centrifugal barrier, so that final states with angular momentum l = 0 are preferred. Again, this does not affect the Trojan horse data. – In the direct method, the proton is a real particle, whereas in the Trojan horse method, the proton transferred from d to 15 N is an “off energy shell” or virtual particle (E ≠ p 2 /(2m)). – The most significant differences are the off- shell effects. The cross section for a resonant reaction (narrow resonance) is given by the well-known Breit-Wigner formula. πλ 2 represents the geometrical cross section. (2J+1)/ (2J 1 +1)(2J 2 +1) is a statistical factor, where J 1 , J 2 and J, represent the angular momenta of the projectile, target and compound nucleus, respectively. Γ a , Γ b are the partial widths of formation and decay, respectively, of the compound nucleus, and Γ = Γ a + Γ b is the total width. E R is the energy of the resonance. Checking the Method on a Simpler Reaction It is beneficial to check a second reaction to see if a similar discrepancy exists there. The reaction 3 He(d,p) 4 He is a good case study, since there is only one significant resonance in this case, and interference effects can be ignored. We first see that the direct and indirect data do not immediately agree. (Upper left below. Red points are indirect data, black points are direct data.) Unfortunately, the data from these two experimental methods (direct and Trojan horse) do not agree, especially at high energy. (This is a graph of S(E) = E e 2πη σ(E), the “astrophysical S-factor”, which removes the exponential term from the cross section. Red points below are Trojan horse data.) 2 2 2 1 2 2 1 () 2 1 2 1 /2 a b R J E J J E E The Breit-Wigner formula cannot correctly reproduce the Trojan horse data, since off-shell effects are involved. We can modify the formula to take these other effects into account. Here φ(p, E) is the wave function of the 16 O nucleus. In the case of 15 N(p,α) 12 C there are multiple resonances involved, as well as subthreshold states, so we have a somewhat more complicated situation in which interference effects are involved. The signs of these interferences are unknown, so several versions of the model are possible. The best model gives the following prediction. Correcting the Trojan Horse Data NASA/JPL-Caltech The data indicates a significantly wider resonance, with a peak at slightly higher energy, than was predicted. Note, by comparison, that the direct data fits the standard R-matrix predictions quite well. (See graph below.) We can apply theoretical corrections to the data as follows: – Find a polynomial fit to the direct data. – Divide this function by the width of the resonance, and multiply by the form factor discussed above. – This gives a prediction for the indirect method. Interestingly, the data still do not fit the predictions and the difference is similar to the case of 15 N(p,α) 12 C. (See upper right graph above—the curve is the prediction based on the direct data.) More work is needed to determine the source of this error. In the mean time, however, the Trojan horse method is still useful in astrophysical studies, since it is reasonably consistent with the direct method at low energies. (See logarithmic plots in the second row above.) It is this region that is of interest in stellar evolution and nucleosynthesis. Special thanks to Dr. Akram Zhanov and the Cyclotron Institute staff. Texas A&M Cyclotron Institute 0.01 0.02 0.05 0.1 0.2 6 8 10 12 14 16 0.01 0.02 0.05 0.1 0.2 6 8 10 12 14 Research supported by: 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1000 2000 3000 4000 2 2 2 1 2 , 2 1 2 1 2 1 /2 b R pE J E J J E E 0.1 0.2 0.3 0.4 0.5 0.6 1000 2000 3000 4000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1000 2000 3000 4000 0.2 0.4 0.6 0.8 2.5 5 7.5 10 12.5 15 17.5 0.1 0.2 0.3 0.4 0.5 2.5 5 7.5 10 12.5 15
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Transcript of Astrophysical S(E)-factor of the 15N(p,α)12C reaction at sub-Coulomb energies via the Trojan-horse...
Astrophysical S(E)-factor of the 15N(p,α)12C reaction at sub-Coulomb energies via the
Trojan-horse methodDaniel Schmidt, Liberty University
Cyclotron Institute —Texas A&M University
Mentor: Dr. Akram Zhanov
Background
The CNO Cycle in Stellar Nucleosynthesis12C + p → 13N + γ13N → 13C + e+ + ν13C + p → 14N + γ14N + p → 15O + γ15O → 15N + e+ + ν15N + p → 12C + α
In this branch of the CNO (carbon, nitrogen, oxygen) cycle, 12C acts as a nuclear catalyst which allows the fusion of four protons into a 4He nucleus (or α particle).
The final step of the CNO cycle, 15N + p → 12C + α (also written 15N(p,α)12C) is the subject of this research.
When a proton impinges on the 15N nucleus, it is repelled by the Coulomb force from the seven other protons, and thus it must have sufficient energy to overcome this “Coulomb barrier” of 2.14 MeV if it is to react. At energies typical of stellar interiors, protons have energies less than 100 keV. The reaction still occurs, but it depends on quantum tunneling.
Accelerator experiments can reproduce the 15N(p,α)12C reaction at E> 70 keV, but at astrophysically relevant energies, the reaction proceeds too slowly to be measured efficiently in this way.
The Trojan Horse Method
An indirect technique, the Trojan horse method, can be used at low energies. In this method, a deuteron impinges on a 15N nucleus. After penetrating the Coulomb barrier, the deuteron breaks, dividing its energy between the proton and neutron. (See diagram below.)
In some cases, the neutron will exit the scene with most of the deuteron’s energy, leaving a low-energy proton inside the Coulomb barrier to react with the 15N nucleus.
By selecting those events in which the neutron leaves with high energy, we can study the reaction at low energies without suppression from the Coulomb barrier.
15N
α
12C
n
16Op
d
The next task is to see if we can find a theoretical correction to the Trojan horse data that will bring the two methods into agreement.
Differences between the two methods include the following:
– In the direct method, the proton is affected by the Coulomb barrier, while in the Trojan horse method it is not.
– The direct data is also affected by a centrifugal barrier, so that final states with angular momentum l = 0 are preferred. Again, this does not affect the Trojan horse data.
– In the direct method, the proton is a real particle, whereas in the Trojan horse method, the proton transferred from d to 15N is an “off energy shell” or virtual particle (E ≠ p2/(2m)).
– The most significant differences are the off-shell effects.
The cross section for a resonant reaction (narrow resonance) is given by the well-known Breit-Wigner formula.
πλ2 represents the geometrical cross section.
(2J+1)/ (2J1+1)(2J2+1) is a statistical factor, where J1, J2 and J, represent the angular momenta of the projectile, target and compound nucleus, respectively.
Γa, Γb are the partial widths of formation and decay, respectively, of the compound nucleus, and Γ = Γa + Γb is the total width.
ER is the energy of the resonance.
Checking the Method on a Simpler Reaction
It is beneficial to check a second reaction to see if a similar discrepancy exists there.
The reaction 3He(d,p)4He is a good case study, since there is only one significant resonance in this case, and interference effects can be ignored.
We first see that the direct and indirect data do not immediately agree. (Upper left below. Red points are indirect data, black points are direct data.)
Unfortunately, the data from these two experimental methods (direct and Trojan horse) do not agree, especially at high energy. (This is a graph of S(E) = E e2πη σ(E), the “astrophysical S-factor”, which removes the exponential term from the cross section. Red points below are Trojan horse data.)
2
2 21 2
2 1( )
2 1 2 1 / 2a b
R
JE
J J E E
The Breit-Wigner formula cannot correctly reproduce the Trojan horse data, since off-shell effects are involved.
We can modify the formula to take these other effects into account.
Here φ(p, E) is the wave function of the 16O nucleus.
In the case of 15N(p,α)12C there are multiple resonances involved, as well as subthreshold states, so we have a somewhat more complicated situation in which interference effects are involved. The signs of these interferences are unknown, so several versions of the model are possible.
The best model gives the following prediction.
Correcting the Trojan Horse Data
NASA/JPL-Caltech
The data indicates a significantly wider resonance, with a peak at slightly higher energy, than was predicted.
Note, by comparison, that the direct data fits the standard R-matrix predictions quite well. (See graph below.)
We can apply theoretical corrections to the data as follows:
– Find a polynomial fit to the direct data.
– Divide this function by the width of the resonance, and multiply by the form factor discussed above.
– This gives a prediction for the indirect method. Interestingly, the data still do not fit the predictions and
the difference is similar to the case of 15N(p,α)12C. (See upper right graph above—the curve is the prediction based on the direct data.)
More work is needed to determine the source of this error. In the mean time, however, the Trojan horse method is still useful in astrophysical studies, since it is reasonably consistent with the direct method at low energies. (See logarithmic plots in the second row above.) It is this region that is of interest in stellar evolution and nucleosynthesis.
Special thanks to Dr. Akram Zhanov and the Cyclotron Institute staff.
Texas A&M Cyclotron Institute
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Research supported by:
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